We begin with simple properties of vectors: addition and scalar multiplication. I assume that you already know how to add two vectors e.g., ${\bf u} + {\bf v}$ and how to multiply a scalar to a vector e.g., $2 {\bf u}$. A little bit of thinking may give you the answer to the following questions: What does $c {\bf u} + d {\bf v}$ represent? or What does $c {\bf u} + d {\bf v} + e {\bf w}$ represent?, where $c, d, e$ are any numbers. These questions may look trivial but introduce you to some deeper concepts in linear systems.

In this lecture, we continue to work on the complexification - combining two complex numbers via transforming one of them into either polar or Cartesian form. After getting the solution of input-output systme using complexification, we investigate some properties of the output as system and input parameters vary. The same framework will be applied to study the input-output system for the 2nd order ODEs. Lastly, we learn how to analyze autonomous ODEs without actually obtaining solutions. The stability of the fixed points plays the central role here.

In this lecture, we consider a 1st-order ODE as an input-output system. This type of ODE is given in a non-homogeneous form. As we have seen in the cooling example, the temperature of a house changes as the external temperature changes. Here, we consider the external temperature as an input and the temperature of the house as an output. This view is quite important because many of real world problems can be seen in this way and our interest is usually to find out the relationship between those two. Next, we learn how to solve non-homogeneous ODEs with sinusoidal forcing using complexification.

This is the second part of analytic solutions where we look into linear ODEs and Bernoulli equations. After studying how to solve these ODEs, we deal with existence and uniqness theorems. Refer to the following table to get a big picture.

Definition

Kullback-Leibler divergence (or distanct) between two probability density functions (PDF) $f$ and $g$ is defined as

Markov inequality

Let $X$ be a non-negative random variable and suppose that $ \mathbb{E} [X] $ exists. For any $ c > 0 $, we have

Analytic solutions mean that solutions can be written in terms of elementary functions such as polynomials, trigonometric function, exponential functions, and hyperbolic functions. There are a few types of first-order ODEs which we can solve analytically. These types are separable, exact, and linear. In the subsequent lectures, we will deal with each of them. It is worth while to note that in the first two cases, the ODE could be nonlinear while in the last case, it should be linear as the name implies. In this world of the first-order ODEs, we will meet the first and the third ones most frequently. We will also meet a few peculiar looking ODEs that don’t look like one of these types at first sight but actually could be reduced to these cases by a certain transformation. (See the following table.)

In the previous class, we looked at a cooling problem. Using a physical principle, we could model this problem as a differential equation which is classified as the first order ordinary differential equations (ODEs). In the first part of this course, we will investigate various types of the first order ODEs and their solution methods.